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@@ -203,5 +203,19 @@ filename "n4.lyx" \end_layout +\begin_layout Chapter +El teorema de Hopf-Rinow +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n5.lyx" + +\end_inset + + +\end_layout + \end_body \end_document diff --git a/ggs/n5.lyx b/ggs/n5.lyx new file mode 100644 index 0000000..f56b96a --- /dev/null +++ b/ggs/n5.lyx @@ -0,0 +1,1360 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +Dada una superficie regular +\begin_inset Formula $S$ +\end_inset + +, un entorno +\begin_inset Formula $W\subseteq S$ +\end_inset + + de +\begin_inset Formula $p\in S$ +\end_inset + + es +\series bold +convexo +\series default + si es normal en todos sus puntos. + Todo +\begin_inset Formula $p\in S$ +\end_inset + + tiene un entorno convexo. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $V$ +\end_inset + + un entorno normal de +\begin_inset Formula $p_{0}\in S$ +\end_inset + +, para cada +\begin_inset Formula $p\in V$ +\end_inset + +, el segmento de geodésica radial +\begin_inset Formula $\gamma_{p}:[0,1]\to V$ +\end_inset + + es el único segmento de geodésica contenido en +\begin_inset Formula $V$ +\end_inset + + con +\begin_inset Formula $\gamma_{p}(0)=p_{0}$ +\end_inset + + y +\begin_inset Formula $\gamma_{p}(1)=p$ +\end_inset + +, salvo reparametrizaciones. + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $\alpha:[a,b]\to V$ +\end_inset + + un segmento de geodésica que une +\begin_inset Formula $p_{0}$ +\end_inset + + a +\begin_inset Formula $p$ +\end_inset + +, por reparametrización afín podemos suponer que +\begin_inset Formula $\alpha:[0,1]\to V$ +\end_inset + +. + Sea +\begin_inset Formula $w:=\alpha'(0)$ +\end_inset + +, la geodésica maximal +\begin_inset Formula $\gamma_{w}:I_{w}\to S$ +\end_inset + + debe cumplir +\begin_inset Formula $[0,1]=\text{Dom}\alpha\subseteq I_{w}$ +\end_inset + +, luego +\begin_inset Formula $1\in I_{w}$ +\end_inset + +, +\begin_inset Formula $w\in{\cal D}_{p}$ +\end_inset + + y +\begin_inset Formula $\alpha(t)=\gamma_{w}(t)=\exp_{p_{0}}(tw)$ +\end_inset + + para +\begin_inset Formula $t\in[0,1]$ +\end_inset + +. + Por otro lado, +\begin_inset Formula $\gamma_{p}(t)=\exp_{p_{0}}(tv_{p})$ +\end_inset + +, y queda probar que +\begin_inset Formula $w=v_{p}$ +\end_inset + +. + Se tiene +\begin_inset Formula $\exp_{p_{0}}(w)=\gamma_{w}(1)=\alpha(1)=p=\gamma_{p}(1)=\exp_{p_{0}}(v_{p})$ +\end_inset + +. + Sea +\begin_inset Formula ${\cal U}$ +\end_inset + + un entorno de +\begin_inset Formula $0_{p_{0}}$ +\end_inset + + tal que +\begin_inset Formula $\exp_{p_{0}}:{\cal U}\to V$ +\end_inset + + es un difeomorfismo, basta ver que +\begin_inset Formula $w\in{\cal U}$ +\end_inset + +, pues entonces, como +\begin_inset Formula $v_{p}\in{\cal U}$ +\end_inset + +, por el difeomorfismo +\begin_inset Formula $w=v_{p}$ +\end_inset + +. + Sean +\begin_inset Formula $\tilde{\alpha}(t):=(\exp_{p_{0}}|_{{\cal U}})^{-1}(\alpha(t))\in{\cal U}$ +\end_inset + + y +\begin_inset Formula $A:=\{t\in[0,1]:\tilde{\alpha}(t)=tw\}$ +\end_inset + +, queremos ver que +\begin_inset Formula $A=[0,1]$ +\end_inset + +. + Como +\begin_inset Formula $\tilde{\alpha}(0)=(\exp_{p_{0}}|_{{\cal U}})^{-1}(p)=0=0w$ +\end_inset + +, +\begin_inset Formula $0\in A$ +\end_inset + +, y +\begin_inset Formula $A=(t\mapsto\tilde{\alpha}(t)-tw)^{-1}(\{0\})$ +\end_inset + + es cerrado. + Ahora bien, para +\begin_inset Formula $t_{0}\in A$ +\end_inset + +, +\begin_inset Formula $t_{0}w=\tilde{\alpha}(t_{0})\in{\cal U}$ +\end_inset + + y, como +\begin_inset Formula ${\cal U}$ +\end_inset + + es abierto, existe +\begin_inset Formula $\varepsilon>0$ +\end_inset + + tal que para +\begin_inset Formula $t\in B(t_{0},\varepsilon)$ +\end_inset + + es +\begin_inset Formula $tw\in{\cal U}$ +\end_inset + + y por tanto +\begin_inset Formula $\alpha(t)=\exp_{p_{0}}(tw)=\exp_{p_{0}}(\tilde{\alpha}(t))$ +\end_inset + +. + Como +\begin_inset Formula $A$ +\end_inset + + es abierto, cerrado y no vacío, +\begin_inset Formula $A=[0,1]$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $\gamma:[0,b)\to S$ +\end_inset + + un segmento de geodésica para el que existe +\begin_inset Formula $\lim_{t\to b^{-}}\gamma(t)=p$ +\end_inset + +, existe +\begin_inset Formula $\varepsilon>0$ +\end_inset + + para el que +\begin_inset Formula $\gamma$ +\end_inset + + se puede extender a una geodésica +\begin_inset Formula $\gamma:[0,b+\varepsilon)\to S$ +\end_inset + + con +\begin_inset Formula $\gamma(b)=p$ +\end_inset + +. + +\series bold +Demostración: +\series default + Existen un entorno convexo +\begin_inset Formula $W$ +\end_inset + + de +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $a\in[0,b)$ +\end_inset + + de modo que +\begin_inset Formula $\gamma(t)\in W$ +\end_inset + + para todo +\begin_inset Formula $t\in[a,b)$ +\end_inset + +. + Dado segmento de geodésica +\begin_inset Formula $\gamma_{p}:[0,1]\to W$ +\end_inset + + que une +\begin_inset Formula $\gamma(a)$ +\end_inset + + a +\begin_inset Formula $p$ +\end_inset + +, para +\begin_inset Formula $t\in[a,b)$ +\end_inset + +, +\begin_inset Formula $\gamma|_{[a,t]}$ +\end_inset + + es una reparametrización de +\begin_inset Formula $\gamma_{p}|_{[0,\frac{t-a}{b-a}]}$ +\end_inset + +, con lo que +\begin_inset Formula $\gamma|_{[a,b)}$ +\end_inset + + es reparametrización de +\begin_inset Formula $\gamma_{p}|_{[0,1)}$ +\end_inset + + y, como +\begin_inset Formula $\gamma_{p}$ +\end_inset + + se existe a un segmento de geodésica +\begin_inset Formula $\gamma_{p}:[0,1+\varepsilon)\to W$ +\end_inset + + para un +\begin_inset Formula $\varepsilon>0$ +\end_inset + +, podemos usar la reparametrización afín para extender +\begin_inset Formula $\gamma$ +\end_inset + + como +\begin_inset Formula $\gamma:[0,b+\frac{\varepsilon}{b-a})\to S$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +Si +\begin_inset Formula $S$ +\end_inset + + es conexa y geodésicamente completa en un +\begin_inset Formula $p\in S$ +\end_inset + +, entonces para todo +\begin_inset Formula $q\in S$ +\end_inset + + existe un segmento de geodésica minimizante que une +\begin_inset Formula $p$ +\end_inset + + a +\begin_inset Formula $q$ +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Para +\begin_inset Formula $p,q\in S$ +\end_inset + +, +\begin_inset Formula $\alpha\in\Omega(p,q)$ +\end_inset + + es +\series bold +minimizante +\series default + o +\series bold +realiza la distancia +\series default + entre +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + + si +\begin_inset Formula $d(p,q)=L(\alpha)$ +\end_inset + +. + Si +\begin_inset Formula $\alpha:[a,b]\to S$ +\end_inset + + realiza la distancia entre +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + +, existe una partición +\begin_inset Formula $a=t_{0}<\dots<t_{n}=b$ +\end_inset + + y un cubrimiento +\begin_inset Formula $\{W_{i}\}_{i=0}^{n}$ +\end_inset + + de +\begin_inset Formula $\alpha([a,b])$ +\end_inset + + por entornos convexos de forma que, para +\begin_inset Formula $i\in\{1,\dots,n\}$ +\end_inset + +, +\begin_inset Formula $\alpha_{i}:=\alpha|_{[t_{i-1},t_{i}]}$ +\end_inset + + es diferenciable e +\begin_inset Formula $\text{Im}\alpha_{i}\subseteq W_{i}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Todo segmento de curva minimizante es una reparametrización de un segmento + de geodésica, y en particular no tiene vértices. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $\alpha:[a,b]\to S$ +\end_inset + + un segmento de curva minimizante y +\begin_inset Formula $a=t_{0}<\dots<t_{n}=b$ +\end_inset + + y +\begin_inset Formula $\{W_{i}\}_{i=0}^{n}$ +\end_inset + + un cubrimiento de +\begin_inset Formula $\alpha([a,b])$ +\end_inset + + por entornos convexos con cada +\begin_inset Formula $\alpha_{i}:=\alpha|_{[t_{i-1},t_{i}]}$ +\end_inset + + diferenciable con imagen en +\begin_inset Formula $W_{i}$ +\end_inset + +, como cada +\begin_inset Formula $\alpha_{i}$ +\end_inset + + es minimizante entre +\begin_inset Formula $\alpha(t_{i-1})$ +\end_inset + + y +\begin_inset Formula $\alpha(t_{i})$ +\end_inset + +, +\begin_inset Formula $L(\alpha_{i})=d(\alpha(t_{i-1}),\alpha(t_{i}))$ +\end_inset + +, pero +\begin_inset Formula $\text{Im}\alpha_{i}\subseteq W_{i}$ +\end_inset + +, luego viendo +\begin_inset Formula $W_{i}$ +\end_inset + + como entorno normal de +\begin_inset Formula $\alpha(t_{i-1})$ +\end_inset + +, +\begin_inset Formula $\alpha_{i}$ +\end_inset + + es una reparametrización de un segmento de geodésica radial de +\begin_inset Formula $\alpha(t_{i-1})$ +\end_inset + + a +\begin_inset Formula $\alpha(t_{i})$ +\end_inset + + y +\begin_inset Formula $\alpha$ +\end_inset + + es una concatenación de geodésicas potenciales. + Además, por continuidad, si +\begin_inset Formula $i<n$ +\end_inset + +, existe +\begin_inset Formula $\varepsilon>0$ +\end_inset + + con +\begin_inset Formula $\alpha([t_{i-1},t_{i}+\varepsilon])\subseteq W$ +\end_inset + + y por tanto un segmento de geodésica radial +\begin_inset Formula $\gamma$ +\end_inset + + en +\begin_inset Formula $W$ +\end_inset + + de +\begin_inset Formula $\alpha(t_{i-1})$ +\end_inset + + a +\begin_inset Formula $\alpha(t_{i}+\varepsilon)$ +\end_inset + +. + Como +\begin_inset Formula $\alpha|_{[t_{i-1},t_{i}+\varepsilon]}$ +\end_inset + + es minimizante, por unicidad es una reparametrización de +\begin_inset Formula $\gamma$ +\end_inset + +, de modo que +\begin_inset Formula $\alpha$ +\end_inset + + es +\begin_inset Formula ${\cal C}^{\infty}$ +\end_inset + + en +\begin_inset Formula $t_{i}$ +\end_inset + + y reparametriza una misma geodésica en todo punto. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $S$ +\end_inset + + es conexa y geodésicamente completa en un +\begin_inset Formula $p_{0}\in S$ +\end_inset + +, entonces para todo +\begin_inset Formula $p\in S$ +\end_inset + + existe un segmento de geodésica minimizante que une +\begin_inset Formula $p_{0}$ +\end_inset + + a +\begin_inset Formula $p$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{TEM} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un espacio topológico +\begin_inset Formula $(X,{\cal T})$ +\end_inset + + es +\series bold +de Hausdorff +\series default + o +\begin_inset Formula $T_{2}$ +\end_inset + + si +\begin_inset Formula $\forall p,q\in X,p\neq q;\exists U\in{\cal E}(p),V\in{\cal E}(q):U\cap V=\emptyset$ +\end_inset + +. + [...] Todo espacio metrizable es [...] +\begin_inset Formula $T_{2}$ +\end_inset + + [...]. + [...] +\end_layout + +\begin_layout Standard +Todo [...] compacto [...] de un espacio [...] Hausdorff [...] es cerrado. + [...] +\end_layout + +\begin_layout Standard +Todo [...] compacto +\begin_inset Formula $K$ +\end_inset + + de un espacio métrico +\begin_inset Formula $(X,d)$ +\end_inset + + es acotado. + +\series bold +Demostración: +\series default + Dado +\begin_inset Formula $a\in X$ +\end_inset + +, para todo +\begin_inset Formula $x\in K$ +\end_inset + + existe un +\begin_inset Formula $n_{x}\in\mathbb{N}$ +\end_inset + + con +\begin_inset Formula $d(x,a)<n_{x}$ +\end_inset + +, de modo que +\begin_inset Formula $\{B(a;n)\}_{n=1}^{\infty}$ +\end_inset + + es un recubrimiento abierto de +\begin_inset Formula $K$ +\end_inset + + del que podemos extraer un subrecubrimiento finito +\begin_inset Formula $\{B(a;n_{1}),\dots,B(a;n_{r})\}$ +\end_inset + +, pero entonces +\begin_inset Formula $K\subseteq B(a;n_{1})\cup\dots\cup B(a;n_{r})=B(a;\max\{n_{1},\dots,n_{r}\})$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Un espacio métrico +\begin_inset Formula $(X,d)$ +\end_inset + + cumple la +\series bold +propiedad de Heine-Borel +\series default + si para +\begin_inset Formula $A\subseteq S$ +\end_inset + +, +\begin_inset Formula $A$ +\end_inset + + es compacto si y sólo si es cerrado y acotado con la distancia +\begin_inset Formula $d$ +\end_inset + +. + Todo espacio métrico que cumple esta propiedad es completo. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $\{p_{n}\}_{n}\subseteq X$ +\end_inset + + una sucesión de Cauchy y +\begin_inset Formula $A:=\{p_{n}\}_{n}$ +\end_inset + + su conjunto de puntos, para +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existe +\begin_inset Formula $N>0$ +\end_inset + + tal que +\begin_inset Formula $\forall n,m\geq N,d(p_{n},p_{m})<\varepsilon$ +\end_inset + +. + Dados +\begin_inset Formula $p\in X$ +\end_inset + + y +\begin_inset Formula $n\geq N$ +\end_inset + +, +\begin_inset Formula $d(p,p_{n})\leq d(p,p_{N})+d(p_{N},p_{n})\leq d(p,p_{N})+\varepsilon$ +\end_inset + +, y dado +\begin_inset Formula $r_{0}>d(p,p_{N})+\varepsilon$ +\end_inset + +, +\begin_inset Formula $d(p,p_{n})<r_{0}$ +\end_inset + + para todo +\begin_inset Formula $n\geq N$ +\end_inset + +. + Tomando +\begin_inset Formula $r:=\max\{r_{0},d(p,p_{1}),\dots,d(p,p_{N-1})\}$ +\end_inset + +, es +\begin_inset Formula $d(p,p_{n})<r$ +\end_inset + + para todo +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +, luego +\begin_inset Formula $A$ +\end_inset + + es acotado. + Por tanto +\begin_inset Formula $\overline{A}$ +\end_inset + + es cerrado y acotado, con lo que +\begin_inset Formula $\overline{A}$ +\end_inset + + es compacto por la propiedad de Heine-Borel y, como +\begin_inset Formula $\{p_{n}\}_{n}\subseteq\overline{A}$ +\end_inset + +, existe una subsucesión convergente de +\begin_inset Formula $(p_{n})_{n}$ +\end_inset + +, pero al ser de Cauchy con una subsucesión convergente, es convergente. +\end_layout + +\begin_layout Standard + +\series bold +Teorema de Hopf-Rinow: +\series default + Dada una superficie regular conexa +\begin_inset Formula $S$ +\end_inset + +, +\begin_inset Formula $(S,d)$ +\end_inset + + es un espacio métrico completo si y sólo si +\begin_inset Formula $S$ +\end_inset + + es geodésicamente completa, si y sólo si existe un +\begin_inset Formula $p_{0}\in S$ +\end_inset + + en el que +\begin_inset Formula $S$ +\end_inset + + es geodésicamente completa, si y sólo si +\begin_inset Formula $(S,d)$ +\end_inset + + cumple la propiedad de Heine-Borel, en cuyo caso diremos que +\begin_inset Formula $S$ +\end_inset + + es +\series bold +completa +\series default +. +\end_layout + +\begin_layout Description +\begin_inset Formula $1\implies2]$ +\end_inset + + Supongamos que +\begin_inset Formula $S$ +\end_inset + + no es geodésicamente completa, con lo que existen +\begin_inset Formula $p_{0}\in S$ +\end_inset + + y +\begin_inset Formula $v\in T_{p_{0}}S$ +\end_inset + + tales que +\begin_inset Formula $\gamma:=\gamma_{v}$ +\end_inset + + no está definida en todo +\begin_inset Formula $t\in\mathbb{R}$ +\end_inset + +. + Podemos suponer que existe +\begin_inset Formula $b>0$ +\end_inset + + tal que +\begin_inset Formula $\gamma$ +\end_inset + + está definida en +\begin_inset Formula $[0,b)$ +\end_inset + + y no se puede extender más allá de +\begin_inset Formula $b$ +\end_inset + +, pues si +\begin_inset Formula $I_{v}$ +\end_inset + + solo estuviera acotado inferiormente, cambiamos +\begin_inset Formula $v$ +\end_inset + + por +\begin_inset Formula $-v$ +\end_inset + +. + Sea +\begin_inset Formula $\{t_{n}\}_{n}\subseteq[0,b)$ +\end_inset + + una sucesión con +\begin_inset Formula $\lim_{n}t_{n}=b$ +\end_inset + +, como +\begin_inset Formula $(t_{n})_{n}$ +\end_inset + + es de convergente es de Cauchy, luego para +\begin_inset Formula $\varepsilon>0$ +\end_inset + + existe +\begin_inset Formula $n_{0}>0$ +\end_inset + + tal que para +\begin_inset Formula $n,m\geq n_{0}$ +\end_inset + + es +\begin_inset Formula $|t_{m}-t_{n}|<\varepsilon$ +\end_inset + +. + Ahora bien, +\begin_inset Formula +\[ +d(\gamma(t_{n}),\gamma(t_{m}))\leq L_{t_{n}}^{t_{m}}(\gamma|_{[t_{n},t_{m}]})=\left|\int_{t_{n}}^{t_{m}}\Vert\gamma'(t)\Vert dt\right|=|t_{m}-t_{n}|\Vert\gamma'(0)\Vert=|t_{m}-t_{n}|\Vert v\Vert, +\] + +\end_inset + +luego si +\begin_inset Formula $n,m\geq n_{0}$ +\end_inset + +, entonces +\begin_inset Formula $d(\gamma(t_{n}),\gamma(t_{m}))\leq\Vert v\Vert\varepsilon$ +\end_inset + +. + Por tanto +\begin_inset Formula $(\gamma(t_{n}))_{n}$ +\end_inset + + es de Cauchy en +\begin_inset Formula $(S,d)$ +\end_inset + + y, como +\begin_inset Formula $(S,d)$ +\end_inset + + es completo, +\begin_inset Formula $(\gamma(t_{n}))_{n}$ +\end_inset + + es convergente, luego existe +\begin_inset Formula $p\in S$ +\end_inset + + con +\begin_inset Formula $p=\lim_{n}\gamma(t_{n})$ +\end_inset + +. + Como +\begin_inset Formula $\{t_{n}\}_{n}$ +\end_inset + + es arbitrario, si +\begin_inset Formula $\{s_{n}\}_{n}\subseteq[0,b)$ +\end_inset + + es otra sucesión con +\begin_inset Formula $\lim_{n}s_{n}=b$ +\end_inset + +, existe +\begin_inset Formula $p'\in S$ +\end_inset + + con +\begin_inset Formula $p'=\lim_{n}\gamma(s_{n})\in S$ +\end_inset + +, y como +\begin_inset Formula +\[ +0\leq d(\gamma(s_{n}),\gamma(t_{n}))\leq L_{s_{n}}^{t_{n}}(\gamma)=\Vert v\Vert|t_{n}-s_{n}|, +\] + +\end_inset + +cuando +\begin_inset Formula $n\to\infty$ +\end_inset + +, +\begin_inset Formula $|t_{n}-s_{n}|\to0$ +\end_inset + + y +\begin_inset Formula $d(p',p)=0$ +\end_inset + +, con lo que +\begin_inset Formula $p'=p$ +\end_inset + + y, como esto se cumple para cualquier +\begin_inset Formula $\{s_{n}\}_{n}$ +\end_inset + +, +\begin_inset Formula $\lim_{t\to b^{-}}\gamma(t)=p$ +\end_inset + +. + Por tanto existe +\begin_inset Formula $\varepsilon>0$ +\end_inset + + tal que +\begin_inset Formula $\gamma$ +\end_inset + + se puede extender a una geodésica +\begin_inset Formula $\gamma:[0,b+\varepsilon)\to S\#$ +\end_inset + +. +\end_layout + +\begin_layout Description +\begin_inset Formula $2\implies3]$ +\end_inset + + Obvio. +\end_layout + +\begin_layout Description +\begin_inset Formula $3\implies4]$ +\end_inset + + Como +\begin_inset Formula $S$ +\end_inset + + es geodésicamente completa en +\begin_inset Formula $p_{0}$ +\end_inset + +, +\begin_inset Formula $\exp_{p_{0}}$ +\end_inset + + está definida en todo +\begin_inset Formula $T_{p_{0}}S$ +\end_inset + +, y queremos ver que, para +\begin_inset Formula $A\subseteq S$ +\end_inset + +, +\begin_inset Formula $A$ +\end_inset + + es compacto si y sólo si es cerrado y acotado. +\end_layout + +\begin_deeper +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Si +\begin_inset Formula $A$ +\end_inset + + es compacto, es cerrado por estar en un espacio Hausdorff y acotado por + estar en uno métrico. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Como +\begin_inset Formula $A$ +\end_inset + + es acotado, existe +\begin_inset Formula $M>0$ +\end_inset + + con +\begin_inset Formula $A\subseteq B_{d}(p_{0},M)$ +\end_inset + +, y como +\begin_inset Formula $S$ +\end_inset + + es conexa y geodésicamente completa en +\begin_inset Formula $p_{0}$ +\end_inset + +, para +\begin_inset Formula $p\in A$ +\end_inset + + existe un segmento de geodésica minimizante +\begin_inset Formula $\gamma:[0,a]\to S$ +\end_inset + + que une +\begin_inset Formula $p_{0}$ +\end_inset + + a +\begin_inset Formula $p$ +\end_inset + +. + Sea +\begin_inset Formula $v:=\gamma'(0)$ +\end_inset + +, entonces +\begin_inset Formula +\[ +M>d(p_{0},p)=L_{0}^{a}(\gamma)=\int_{0}^{a}\Vert\gamma'(t)\Vert dt=a\Vert\gamma'(0)\Vert=a\Vert v\Vert, +\] + +\end_inset + +luego +\begin_inset Formula $av\in{\cal D}(0,M)$ +\end_inset + + y +\begin_inset Formula $p=\gamma(a)=\exp_{p_{0}}(av)\in\exp_{p_{0}}({\cal D}(0,M))\subseteq\exp_{p_{0}}(\overline{{\cal D}(0,M)})$ +\end_inset + +. + Pero +\begin_inset Formula $\exp_{p_{0}}(\overline{{\cal D}(0,M)})$ +\end_inset + + es compacto en +\begin_inset Formula $S$ +\end_inset + + por serlo +\begin_inset Formula $\overline{{\cal D}(0,M)}$ +\end_inset + + en +\begin_inset Formula $T_{p_{0}}S$ +\end_inset + + y por ser +\begin_inset Formula $\exp_{p_{0}}$ +\end_inset + + continua, luego +\begin_inset Formula $A\subseteq\exp_{p_{0}}(\overline{{\cal D}(0,M)})$ +\end_inset + + es un cerrado dentro de un compacto y por tanto un compacto. +\end_layout + +\end_deeper +\begin_layout Description +\begin_inset Formula $4\implies1]$ +\end_inset + + Visto para todo espacio métrico. +\end_layout + +\begin_layout Standard +Así, si una superficie regular conexa +\begin_inset Formula $S$ +\end_inset + + es completa, dos puntos +\begin_inset Formula $p,q\in S$ +\end_inset + + se pueden unir con un segmento de geodésica minimizante, no necesariamente + único. + Todo espacio métrico compacto es completo, pues sus subespacios cerrados + y acotados, por ser cerrados, son compactos, cumpliendo la propiedad de + Heine-Borel. + En particular toda superficie regular, conexa y compacta es completa. +\end_layout + +\begin_layout Standard +Toda superficie regular conexa y cerrada en +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + + es completa. + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $S$ +\end_inset + + esta superficie, dada una sucesión de Cauchy +\begin_inset Formula $\{p_{n}\}_{n}\subseteq S$ +\end_inset + +, como +\begin_inset Formula $\Vert p_{n}-p_{m}\Vert\leq d(p_{n},p_{m})$ +\end_inset + + para +\begin_inset Formula $n,m\in\mathbb{N}$ +\end_inset + +, +\begin_inset Formula $(p_{n})_{n}$ +\end_inset + + también es una sucesión de Cauchy en +\begin_inset Formula $(\mathbb{R}^{3},\Vert\cdot\Vert)$ +\end_inset + +, pero este espacio es completo y por tanto +\begin_inset Formula $(p_{n})_{n}$ +\end_inset + + converge en +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + + a un +\begin_inset Formula $p\in\mathbb{R}^{3}$ +\end_inset + +. + Pero como +\begin_inset Formula $S$ +\end_inset + + es cerrada y +\begin_inset Formula $\{p_{n}\}_{n}\subseteq S$ +\end_inset + +, el límite +\begin_inset Formula $p\in S$ +\end_inset + +, luego la sucesión converge también en +\begin_inset Formula $S$ +\end_inset + +. +\end_layout + +\end_body +\end_document |